Optimization device, method for controlling optimization processing, and non-transitory computer-readable storage medium for storing program for controlling optimization processing

ABSTRACT

A method for controlling an optimization device, the method comprising: obtaining a value of a state variable included in an evaluation function obtained by converting a problem stored in a storage unit; and performing a search for a minimum value of the evaluation function by repeating a process of updating the value of the state variable by a Markov chain Monte Carlo method based on a transition probability distribution that is represented by a function by which it is possible to normalize to one a sum of probabilities of transition from a current state represented by a current value of the state variable to each of a plurality of different states, in which a transition probability becomes larger than in a Boltzmann distribution as a change in the value of the evaluation function due to a change in the value of the state variable is positively larger.

CROSS-REFERENCE TO RELATED APPLICATION

This application is based upon and claims the benefit of priority of theprior Japanese Patent Application No. 2019-112367, filed on Jun. 17,2019, the entire contents of which are incorporated herein by reference.

FIELD

The embodiments discussed herein are related to an optimization device,a method for controlling an optimization processing, and anon-transitory computer-readable storage medium storing a program forcontrolling the optimization processing.

BACKGROUND

A problem that frequently occurs in natural sciences and social sciencesis a minimum value solution problem (or also called a combinatorialoptimization problem) that finds the minimum value of an evaluationfunction (or also called an energy function) (or that finds acombination of values of state variables of an evaluation function thatgives the minimum value). In recent years, movement to formulate such aproblem with an Ising model, which is a model representing the behaviorsof spins of a magnetic material, has been accelerated. A technical basisof this movement is realization of an Ising-type quantum computer. TheIsing-type quantum computer is expected to be capable of solving, in arealistic time, multivariable combinatorial optimization problems thatNeumann-type computers are inapt at. On the other hand, an optimizationdevice in which an Ising-type computer is implemented by an electroniccircuit has been developed.

As a calculation method for the minimum value solution problem using anIsing model, there is a method of searching for a minimum value of anIsing-type energy function using a Markov chain Monte Carlo method(hereinafter, referred to as an MCMC method or an MCMC calculation). Inthe MCMC calculation, it is common to update state variables of theenergy function (state transition) with a transition probabilityaccording to a Boltzmann distribution. However, the energy functionincludes many local minima that are local solutions, and once a solutionis captured by a local solution, a probability of escape from the localsolution is very low. To facilitate escape from the local solution,techniques such as a simulated annealing method (hereinafter abbreviatedas SA method) and a replica exchange method (also called an extendedensemble method or the like) are known.

The SA method is a method such that the value of a temperature parameterincluded in an equation that determines the transition probability isgradually reduced according to a predetermined schedule (which isequivalent to lowering the temperature), to thereby finally obtain theminimum value of the energy function. The replica exchange method is atechnique to set a different value of a temperature parameter for eachof a plurality of replicas, perform the MCMC calculation independentlyfor each replica, and exchange values of temperature parameters (orstate variables) between replicas with a predetermined exchangeprobability.

However, in the SA method, since the minimum value may not be obtainedunless the value of the temperature parameter is reduced very slowly,the calculation efficiency is poor. Furthermore, the replica exchangemethod may not efficiently search for the minimum value unless the valueof the temperature parameter set to each replica moves all the way froma low temperature region to a high temperature region, but setting ofthe temperature parameter for that purpose is difficult. On the otherhand, when the solution falls into a local minimum, there is a methodfor facilitating escape from the local minimum by adding an offset toenergy included in the equation that determines the transitionprobability to increase the transition probability in a direction toincrease the energy (hereinafter called a dynamic offset method).

Examples of the related art include Japanese Laid-open PatentPublication No. 2018-5541, and Japanese Patent No. 6465231.

Examples of the related art also include Sanroku Tsukamoto, MotomuTakatsu, Satoshi Matsubara and Hirotaka Tamura, “An AcceleratorArchitecture for Combinatorial Optimization Problems”, FUJITSU Sci.Tech. J., Vol. 53, No. 5, September, 2017, pp. 8-13, S. Kirkpatrick, C.D. Gelatt Jr, M. P. Vecchi, “Optimization by Simulated Annealing”,Science, Vol. 220, No. 4598, 13 May, 1983, pp. 671-680, ConstantinoTsallis, Daniel A. Stariolo, “Generalized simulated annealing”, PhysicaA, 233, 1996, pp. 395-406, and Koji Hukushima and Koji Nemoto, “ExchangeMonte Carlo Method and Application to Spin Glass Simulations”, J. Phys.Soc. Jpn, Vol. 65, No. 6, June, 1996, pp. 1604-1608.

SUMMARY

According to an aspect of the embodiments, an optimization deviceincludes: a memory configured to store a value of a state variableincluded in an evaluation function, the evaluation function beingobtained by converting a problem; and a processor coupled to the memory,the processor being configured to perform a search for a minimum valueof the evaluation function by repeating a process of updating the valueof the state variable by a Markov chain Monte Carlo method based on afirst transition probability distribution, the first transitionprobability distribution being a transition probability distributionthat is represented by a first function, the first function being afunction configured to normalize to one a sum of probabilities oftransition from a current state to each of a plurality of differentstates, the current state being represented by a current value of thestate variable, the first transition probability distribution beingconfigured to have a feature that a transition probability becomeslarger than in a Boltzmann distribution as a change in the value of theevaluation function due to a change in the value of the state variableis positively larger.

The object and advantages of the invention will be realized and attainedby means of the elements and combinations particularly pointed out inthe claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and arenot restrictive of the invention.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram illustrating an example of an optimization device ofa first embodiment;

FIG. 2 is a diagram illustrating an example of comparing a Boltzmanndistribution with two distributions different from the Boltzmanndistribution;

FIG. 3 is a diagram illustrating an example of hardware of anoptimization device of a second embodiment;

FIG. 4 is a block diagram illustrating a function example of anoptimization device of a second embodiment;

FIG. 5 is a flowchart illustrating a process flow of an example of amethod for controlling the optimization device of the second embodiment;

FIG. 6 is a flowchart illustrating a process flow of an example of aninformation reading process;

FIG. 7 is a flowchart illustrating a process flow of an example of aninitialization process;

FIG. 8 is a flowchart illustrating a process flow of an example of anM-H calculation process;

FIG. 9 is a flowchart illustrating a process flow of an example of anenergy update process;

FIG. 10 is a flowchart illustrating a process flow of an example of anannealing process;

FIG. 11 is a flowchart illustrating a process flow of an example of anSA process;

FIG. 12 is a flowchart illustrating a process flow of an example of a λannealing process;

FIG. 13 is a flowchart illustrating a process flow of an example of areplica exchange process;

FIG. 14 is a diagram illustrating an example of a method for determininga replica number of a pair for which replica exchange is performed;

FIG. 15 is a flowchart illustrating a process flow of an example of asample output process;

FIG. 16 is a diagram illustrating an example of comparing samplingresults when two types of transition probability distributions are used;

FIG. 17 is a diagram illustrating an example of comparing samplingresults when two types of transition probability distributions withtemperatures being changed are used;

FIG. 18 is a flowchart illustrating a process flow of an example of aparameter optimization process;

FIG. 19 is a diagram illustrating an output example of evaluationvalues;

FIG. 20 is a diagram illustrating an example of a stationary statecalculation result when three types of transition probabilitydistributions are used;

FIG. 21 is a diagram illustrating an example of comparing searchperformance;

FIG. 22 is a diagram illustrating an example of calculation results oftemperature transitions in respective replicas when the Boltzmanndistribution is used;

FIG. 23 is a diagram illustrating an example of calculation results ofenergy in the respective replicas when the Boltzmann distribution isused;

FIG. 24 is a diagram illustrating an example of calculation results oftemperature transitions in the respective replicas when a power law-typetransition probability distribution is used;

FIG. 25 is a diagram illustrating an example of calculation results ofenergy in the respective replicas when the power law-type transitionprobability distribution is used;

FIG. 26 is a diagram illustrating an example of calculation results ofthe replica exchange method when the Boltzmann distribution and thepower law-type transition probability distribution are used;

FIG. 27 is an enlarged view of regions where energy is small in thecalculation results of FIG. 26 ;

FIG. 28 is a diagram illustrating an example of calculation results ofthe λ annealing method;

FIG. 29 is a diagram illustrating an example of comparing a calculationresult by a device that performs a parallel trial using a digitalcircuit with a calculation result by the optimization device of thesecond embodiment; and

FIG. 30 is a diagram illustrating an example of comparing searchperformance for an energy function that is a continuous function.

DESCRIPTION OF EMBODIMENT(S)

However, when the dynamic offset method is used to facilitate escape ofthe solution from the local minimum and efficiently search for theminimum value of the evaluation function, an offset is added to theenergy and the Markov chain is destroyed, and thus accuracy of thesolution may be degraded. Furthermore, even when the solution onceescapes from the local minimum, it may be captured again by the samelocal minimum.

According to an aspect of the embodiments, provided are an optimizationdevice, a method for controlling the optimization device, and a programfor controlling the optimization device that can efficiently search fora minimum value of an evaluation function without breaking a Markovchain.

In one aspect, it is possible to efficiently search for a minimum valueof an evaluation function without breaking the Markov chain.

Hereinafter, embodiments will be described with reference to thedrawings.

First Embodiment

FIG. 1 is a diagram illustrating an example of an optimization device ofa first embodiment.

A storage unit 11 stores a value of a state variable included in anevaluation function (hereinafter, referred to as an energy function)obtained by converting a problem, a value of the evaluation functioncorresponding to the value of the state variable (hereinafter, referredto as energy), and the like. The storage unit 11 is a volatile storagedevice such as a random access memory (RAM) or a non-volatile storagedevice such as a hard disk drive (HDD) or a flash memory.

A processing unit 12 searches for a minimum value of the energy functionby repeating a process of updating the value of the state variable by anMCMC method. The processing unit 12 is a processor such as a centralprocessing unit (CPU), a general-purpose computing on graphicsprocessing units (GPGPU), or a digital signal processor (DSP). However,the processing unit 12 may include an electronic circuit for a specificapplication such as an application specific integrated circuit (ASIC), afield programmable gate array (FPGA), and the like. The processorexecutes a program stored in a memory such as a RAM. For example, aprogram for controlling the optimization device 10 is executed. Notethat a set of a plurality of processors are sometimes referred to as a“multiprocessor” or simply a “processor”.

The optimization device 10 searches for, for example, a minimum value(or a combination of values of state variables that can obtain theminimum value) of an Ising-type energy function obtained by convertingthe problem.

The Ising-type energy function (H({x})) is defined, for example, by thefollowing equation (1).H({x})=−Σ_(i=1) ^(N)Σ_(j>i) ^(N) W _(ij) x _(i) x _(j)−Σ_(i=1) ^(N) b_(i) x _(i) −C  (1)

The first term on the right side is an accumulation of a product ofvalues (0 or 1) of two state variables and a weighting factor for allcombinations of N state variables without omission and duplication.x_(i) represents an i-th state variable, x_(j) represents a j-th statevariable, and W_(ij) is a weighting factor indicating the magnitude ofan interaction between x_(i) and x_(j). The second term on the rightside is a total sum of products of bias coefficients (b_(i)) obtainedfor respective state variables and the values of the respective statevariables, and the third term (C) on the right side is a constant.

Furthermore, a change in the value of the energy function (energydifference (ΔE_(k))) due to a change in the value of x_(k), which is oneof the state variables, can be expressed as ΔE_(k)=−(1−2x_(k))h_(k).1−2x_(k) represents a variation (Δx_(k)) of x_(k). When x_(k) changesfrom 1 to 0, 1−2x_(k)=−1, and when x_(k) changes from 0 to 1,1−2x_(k)=1. Furthermore, h_(k) is called a local field and can beexpressed by the following equation (2). Noted that the equation (2) isrepresented in an abbreviated form for illustrative purposes. Forexample, the right side of the equation (2) may be added with otherparameters, such as a value regarding a penalty factor of TravellingSalesman Problem (TSP).h _(k)=Σ_(i=1,i≠k) ^(N) W _(ki) x _(i)  (2)

When searching for the minimum value of the energy function as describedabove, the processing unit 12 repeats the process of updating the valueof any one of a plurality of state variables by an MCMC calculation. Astochastic process underlying the MCMC calculation theory is describedby the following equation (3).π_(j)({x},t+Δt)−π_(j)({x},t)=ΔtΣ _(i≠j) ^(∞)(π_(i)({x},t)P_(i→j)−π_(j)({x},t)P _(j→i))  (3)

In equation (3), π_(i)({x}, t) is a probability that a state defined by{x}(indicating the value of each state variable) is realized. P_(i→j)represents a transition probability of transition from state i to statej. In the MCMC calculation, it is guaranteed that a stationarydistribution is achieved by repeating the calculation, and thestationary state is described by the following equation (4).0=Δt Σ _(i≠j) ^(∞)(π_(i)({x},t)P _(i→j)−π_(j)({x},t)P _(j→i))  (4)

Here, π_(i)({x}, t) does not depend on time when a Markov chain is in astationary state. Therefore, equation (4) is a general equation, but inequation (4), a condition in which an equation holds for each term of asum is often used. This condition is called the principle of detailedbalance and is expressed by the following equation (5).0=π_(i)({x},t)P _(i→j)−π_(j)({x},t)P _(j→i)  (5)

Next, a Metropolis-Hastings method (hereinafter abbreviated as the M-Hmethod) which is a core part of the MCMC calculation will be described.

In the M-H method, the following equation (6) is used as a transitionprobability that satisfies the principle of detailed balance (equation(5)).

$\begin{matrix}{P_{i->j} = {\min\left( {1,\frac{\pi_{j}\left( \left\{ x \right\} \right)}{\pi_{i}\left( \left\{ x \right\} \right)}} \right)}} & (6)\end{matrix}$

When an energy difference (ΔE=E_(j)−E_(i)=H({x_(j)})−H({x_(i)})) at thetime of a state transition from a state i to a state j is smaller than0, it is a state transition in which energy decreases, and thus equation(6) becomes P_(i→j)=1. On the other hand, since an opposite statetransition is a state transition in which energy increases,P_(j→i)=π_(i)({x})/π_(j) ({x}). When these are substituted into equation(5), the following equation (7) is satisfied, which satisfies theprinciple of detailed balance.

$\begin{matrix}{{{{\pi_{i}\left( \left\{ x \right\} \right)}1} - {{\pi_{j}\left( \left\{ x \right\} \right)}\frac{\pi_{i}\left( \left\{ x \right\} \right)}{\pi_{j}\left( \left\{ x \right\} \right)}}} = 0} & (7)\end{matrix}$

Conversely, also when ΔE=E_(j)−E_(i)≥0, the principle of detailedbalance is satisfied by a similar logic. Therefore, since the Markovchain converges uniquely, a sampled probability distribution isπ_(i)({x}).

In the M-H method, when a Boltzmann distribution is used as aprobability distribution of π({x}, t) as conventionally used, it can beexpressed as π({x}, t)=exp(−βE) (β is an inverse temperature (reciprocalof the value of a temperature parameter)). Then, equation (6) can beexpressed as the following equation (8).

$\begin{matrix}{P_{i->j} = {{\min\left( {1,\frac{\pi_{j}\left( \left\{ x \right\} \right)}{\pi_{i}\left( \left\{ x \right\} \right)}} \right)} = {\min\left( {1,{\exp\left( {{- \beta}\;\Delta\; E} \right)}} \right)}}} & (8)\end{matrix}$

However, when the Boltzmann distribution is used as described above,once a solution is captured by a local solution, a probability of escapefrom the local solution becomes very low. This is because when thetransition probability is represented by equation (8), a probability ofexponentially exceeding an energy barrier decreases as ΔE increasespositively. For example, even when −βΔE=approximately 20, sinceexp(−20)≈2.06×10⁻⁹, the state transition of −βΔE=20 occurs only aboutonce in 2.1 billion calculations (the number of MCMC calculations).

For this reason, there is a method for facilitating escape from thelocal solution using the SA method, the replica exchange method, or thedynamic offset method, but there is a technical problem as describedabove.

When dealing with a problem on a physical system such as a magneticmaterial, the Boltzmann distribution has to be used. This is because athermal equilibrium state follows a statistical distribution ofexp(−βE). This is because a simulation of dynamic behaviors of a systemthat does not satisfy the laws of physics is meaningless. Furthermore,when an entropy effect plays an important role in thermodynamicbehaviors of a system, such as Helmholtz free energy or Gibbs freeenergy, an available transition probability distribution is alsorestricted in order to properly incorporate the entropy effect.

However, the problem of finding the minimum value of the energy functionsuch as equation (1) represented by the Ising model just needs to besimply regarded as a minimum value solution problem of a function, andthus does not have to be bound to the Boltzmann distribution.

Note that in the energy function such as equation (1), there are 2^(N)solutions. A set {E} of them arranged in ascending order of energy is{E}={E₀, E₁, E₂, . . . , E_(i) . . . , E_(j), . . . , E_(M)}. Forexample, if i<j, E_(i)<E_(j). Here, M is M≤2^(N). This is becausesolutions are allowed to overlap. Then, E₀ is bounded below. Forexample, it does not diverge to negative infinity, and there is aminimum value.

Hereinafter, before presenting a distribution that replaces theBoltzmann distribution, first, the transition probability is defined asin the following equation (9).P _(i→j)=min(1,f(ΔE))  (9)

In equation (9), f(ΔE) is a function that is 1 when ΔE<0, or that anequation of f(ΔE) is applied when ΔE≥0. Furthermore, a normalizedintegral of f(ΔE) exists. The existence of a normalized integral meansthat the normalized integral is not zero and does not diverge topositive infinity or negative infinity, but becomes a finite value(positive real number). For example, f(ΔE) such that the value of thenormalized integral becomes 1 as in the following equation (10) can beselected.∫₀ ^(∞) f(ΔE)d{ΔE}=1  (10)

For example, f(ΔE) is represented by a function that can be normalizedso that the sum of probabilities (transition probabilities) of atransition from the current state represented by the value of thecurrent state variable to each of a plurality of different states(transition probability) becomes 1. However, f(ΔE) is selected so thatf(0)=1.

A stochastic matrix is defined as in the following equation (11).

$\begin{matrix}{P = \begin{pmatrix}p_{11} & p_{12} & \ldots & p_{1M} \\p_{21} & p_{22} & \ldots & p_{2M} \\\ldots & \ldots & \ldots & \ldots \\p_{M - 11} & p_{M - 12} & \ldots & p_{M - {1M}} \\p_{M\; 1} & p_{M\; 2} & \ldots & p_{MM}\end{pmatrix}} & (11)\end{matrix}$

Furthermore, each element of the stochastic matrix (P) is defined as inthe following equation (12).

$\begin{matrix}{{p_{ij} = \frac{f\left( {\Delta\; E_{ij}} \right)}{Z}}{Z = {\sum\limits_{j = 1}^{M}{f\left( {\Delta\; E_{ij}} \right)}}}} & (12)\end{matrix}$

In equation (12), ΔE_(ij)=E_(j)−E_(i). Therefore, the following equation(13) holds for each row of the stochastic matrix.Σ_(j=1) ^(M) p _(ij)=1  (13)

Therefore, this stochastic matrix has an eigenvalue of 1 and forms astationary Markov chain.

Note that equations (9) to (13) described above are formulated for thecase where the state variable is a value of 0 or 1 (when the energy isdiscrete) as in the Ising-type evaluation function, but may also beapplied to a case where the state variable is a continuous variable(when the energy is a continuous function). The reason will be brieflydescribed below. When a continuous variable is used, a transitionprobability in E_(i) is used assuming that a transition probability in asmall section [E_(i), E_(i)+ΔE] does not change for an energy section[E₀, E₁]. Then, if a transition probability from a state of [E_(i),E_(i)+ΔE] to [E_(j), E_(j)+ΔE] is f(ΔE)=f(E_(j)−E_(j)), it becomesexactly the same calculation as that when the Ising-type evaluationfunction that is a discrete model is used. Note that since a basic partof the proof is the same, the proof for a case where the state variableis a continuous variable is omitted.

Note that the method described above generally breaks the principle ofdetailed balance represented in equation (5). When the Boltzmanndistribution, which is a special case of this method, is used (whenf(ΔE)=exp(−βΔE)), the principle of detailed balance is satisfied.

Next, an example of a transition probability distribution (f(ΔE)) thatreplaces the Boltzmann distribution will be described. A conditionneeded for f(ΔE) is a condition that it can be normalized as expressedin equation (10), and any transition probability distribution may beapplied if this condition is satisfied. However, the processing unit 12of the optimization device 10 uses transition probability distributionsrepresented by, for example, the following equations (14), (15), and(16) from the viewpoint of solving the minimum value solution problem(optimization problem) efficiently.

$\begin{matrix}{{f\left( {\Delta\; E} \right)} = \frac{1}{\left( {1 + {\beta\;\Delta\; E}} \right)^{m_{1}}}} & (14) \\{{f\left( {\Delta\; E} \right)} = \frac{1}{\left( {1 + \left( {\beta\;\Delta\; E} \right)^{2}} \right)^{m_{2}/2}}} & (15) \\{{f\left( {\Delta\; E} \right)} = {\exp\left\{ {- \left( {{\beta\Delta}\; E} \right)^{m_{3}}} \right\}}} & (16)\end{matrix}$

However, m₁>1, m₂>1, and m₃>0 in equations (14) to (16) in order tosatisfy the normalization condition of equation (10).

Note that Non Patent Literature 3 describes a transition probabilitydistribution that can be expressed asP_(q)(x_(t)→x_(t+1))=1/[1+(q−1)βΔE]^(1/q−1) (refer to equation (5) inNon Patent Literature 3). This equation is different from equation (14).This is because when q=2, the result is the same as when m₁=1 inequation (14), but in equation (14), m₁>1 as described above. Note thatin Non Patent Literature 3, when q≥2 in 1/[1+(q−1)βΔE]^(1/q−1), thenormalized integral diverges to infinity (there is no normalizedintegral), and thus the transition probability becomes stochasticallyincorrect. Furthermore, in Non Patent Literature 3 when q=1, thetransition probability distribution becomes the Boltzmann distribution.When q<1, the transition probability is represented by a power. Forexample, when q=½, it becomes P_(q)(x_(t)−x_(t+1))=(1−βΔE/2)², but when1−βΔE/2<0 is satisfied, which is when ΔE becomes positively large, thetransition probability diverges to infinity. For this reason, in NonPatent Literature 3 under such conditions, the transition probability isset to 0, but in this case, when the solution is captured by a localminimum with a deep value (large ΔE), it becomes impossible to escape.Equation (14) has been formulated so that these situations do not occur.

Note that the transition probability distribution represented byequation (15) is called Cauchy distribution when m₂=2. The transitionprobability distribution represented by equation (16) is a Boltzmanndistribution when m₃=1, and is a normal distribution when m₃=2.

Therefore, among equations (14) to (16), a distribution in a case otherthan m₃=1 in equation (16) is the transition probability distributionthat replaces the Boltzmann distribution.

Note that the transition probability distribution to be used just needsto be a distribution in which the greater the energy difference, thegreater the transition probability than in the Boltzmann distribution,but it is desirable to selectively use a transition probabilitydistribution depending on the application. For example, in terms of theamount of calculation, in equations (14) and (15), the transitionprobability distribution approximately becomes a function to be definedby m, such as f(ΔE)≈(βΔE)^(−m) in the region of ΔE>>1. When m=1+δ, bytaking δ sufficiently small, the transition probability distributionasymptotically becomes like the following equation (17).

$\begin{matrix}{{{f\left( {\Delta\; E} \right)} \cong {\lim\limits_{\delta->0}\left( {\beta\;\Delta\; E} \right)^{- {({1 + \delta})}}}} = \left( {\beta\;\Delta\; E} \right)^{- 1}} & (17)\end{matrix}$

From the viewpoint of the amount of calculation, when the Boltzmanndistribution is used as the transition probability distribution,N_(B)=exp(βΔE), where N_(B) is an average number of calculationsexceeding ΔE. On the other hand, when equations (14) and (15) are used,a relationship represented by N_(f)≈(βΔE)^(m) holds, where N_(f) is theaverage calculation number exceeding ΔE, and then letting the solutionescape from the local minimum of the energy function may be consideredto be merely a polynomial time (P) problem of the calculation amountdefined by a power. Noted that N_(f) may be referred to as an averagenumber of calculations exceeding ΔE.

As described above, the processing unit 12 may solve the problem thatthe transition probability decreases exponentially when ΔE is large byusing the transition probability distribution in which the transitionprobability is larger than in the Boltzmann distribution as ΔE islarger.

Next, it is considered how to improve efficiency of energy searchcapability when solving the minimum value solution problem.

In the M-H method, a state transition satisfying ΔE<0 is accepted, and astate transition in which energy is increased is also acceptedstochastically. Due to the nature of the M-H method, when the statetransition in the region of ΔE≈0 is accepted too much, directions inwhich the energy increases and directions in which the energy decreasesare equalized. As a result, the energy search capability is reduced.

When the Boltzmann distribution is used as the transition probabilitydistribution, since a derivative of the transition probability changesdiscontinuously at the boundary of ΔE=0, a state transition with a smallincrease in energy is efficiently rejected. This is an advantage whenusing the Boltzmann distribution as the transition probabilitydistribution.

The processing unit 12 uses this advantage of the Boltzmanndistribution, and makes the transition probability proportional to thepower of DE in the region of ΔE>>1. Accordingly, when using equation(14), the processing unit 12 just needs to set m₁=1+δ, and set δ>0 andδ<<1. For example, m₁=1.0001 just needs to be set.

FIG. 2 is a diagram illustrating an example of comparing a Boltzmanndistribution with two distributions different from the Boltzmanndistribution. In FIG. 2 , the horizontal axis represents x (=βΔE), andthe vertical axis represents f(x) (transition probability). Ranges of xand f(x) are different between a left graph and a right graph in FIG. 2.

In FIG. 2 , a distribution (Boltzmann distribution) represented byf(x)=exp(−x) and two distributions represented by f(x)=1/(1+x²) andf(x)=1/(1+x) are illustrated. In the transition probability distributionrepresented in equation (14), by making m₁ close to 1 as describedabove, it becomes almost the same distribution as f(x)=1/(1+x).f(x)=1/(1+x²) is equal to a transition probability distribution wherem₂=2 in equation (15).

As can be seen from the graph on the left side of FIG. 2 , f(x)=1/(1+x)asymptotically approaches the Boltzmann distribution in a region whereΔE≈0. Furthermore, as can be seen from the graph on the right side inFIG. 2 , 1/(1+x) and 1/(1+x²) are values (transition probabilities)larger than exp(−x) in the region of x (=βΔE)>>1. When βΔE=20, f(x) isapproximately exp(−20.0)≈2.1×10⁻⁹ in the Boltzmann distribution.Therefore, the state transition in which βΔE=20 is accepted only aboutonce every 2.1 billion times. On the other hand, in the transitionprobability distribution represented by equation (14), when m₁ is madeclose to 1 and approximated to 1/(1+x), the transition probability isabout 0.05 when ΔE=20, and thus the state transition in which βΔE=20 isaccepted once every 20 times. Therefore, the state transition can begenerated 24 million times more efficiently than when the Boltzmanndistribution is used.

FIG. 1 illustrates an example of a method for controlling theoptimization device 10.

The processing unit 12 obtains problem information (step S1). Theproblem information includes, for example, a weighting factor (W_(ij)),a bias coefficient (b_(i)), a constant (C), and the like of the energyfunction represented in equation (1). The problem information mayinclude information of a transition probability distribution to be used(including values of m₁ to m₃ in equations (14) to (16) describedabove), temperature information (for example, an inverse temperature(β)), a value of the state variable included in the energy function, thenumber of calculations as an end condition for the MCMC calculation, andthe like. The problem information may be supplied from the outside ormay be stored in the storage unit 11 in advance.

Next, the processing unit 12 performs an initialization process (stepS2). The initialization process includes a process of initializing thestate variables x₁ to x_(N) stored in the storage unit 11 when theenergy function is represented by equation (1). For example, x₁ to x_(N)may be all initialized to 0, or may be all initialized to 1. Further, x₁to x_(N) may be initialized such that 0 and 1 are set at random, or maybe initialized by a value supplied from the outside. Furthermore, theinitialization process includes a process of calculating an initialvalue of energy by equation (1) based on the problem information andinitial values of the state variables. The initial value of energy isstored in the storage unit 11 as a current minimum value (E_(min)).

Thereafter, the processing unit 12 performs an update process (step S3).In the update process, the processing unit 12 updates the values of thestate variables by the MCMC calculation based on a transitionprobability distribution that satisfies the above-describednormalization condition and has a larger transition probability than theBoltzmann distribution as ΔE is larger, and is represented, for example,by any of the above-described equations (14) to (16).

For example, the processing unit 12 selects one of x₁ to x_(N) at randomand calculates ΔE when the value of the selected state variable isinverted (changed from 0 to 1 or 1 to 0). Then, if ΔE is negative, theprocessing unit 12 inverts the value of the selected state variable(determines the update). If ΔE is positive, for example, if f(ΔE)<R istrue based on a comparison result of f(ΔE) in any of equations (14) to(16) with a uniform random number R of 0≤R≤1, the processing unit 12inverts the value of the selected state variable. If f(ΔE)<R is nottrue, the processing unit 12 does not invert the value of the selectedstate variable. Noted that, in view of the application to developing ofa concrete system, the above relationship between f(ΔE) and the randomnumber R may be reversed. For example, the relationship may be adjustedsuch that if f(ΔE)>R is true then the processing unit 12 inverts thevalue of the selected state variable.

Furthermore, in the process of step S3, when the value of the statevariable is inverted, the processing unit 12 updates the energy. Whenthe updated energy is the minimum value so far, the processing unit 12updates E_(min) stored in the storage unit 11. Note that the storageunit 11 may store the value of each state variable when E_(min) isobtained as a candidate for an optimal solution.

After the processing in step S3, the processing unit 12 determineswhether or not the number of MCMC calculations has reached apredetermined number (N_(E)) that is an end condition (step S4). Whenthe number of calculations has not reached N_(E), the process from stepS3 is repeated. When the number of calculations has reached N_(E), theprocessing unit 12 outputs the current E_(min) stored in the storageunit 11 as a current calculation result to, for example, an externaldevice (an external computer, a storage medium, a display device, or thelike) (step S5), and ends the process.

With the optimization device 10 of the first embodiment as describedabove, the transition probability distribution in which the transitionprobability is larger than in the Boltzmann distribution as ΔE ispositively larger is applied, and thus a solution can escape from alocal solution with high efficiency. Furthermore, since it is not amethod of adding an offset to energy as in the dynamic offset method,the Markov chain may not be broken.

From the above, it is possible to efficiently search for a minimum valueof an energy function without breaking the Markov chain.

Second Embodiment

FIG. 3 is a diagram illustrating an example of hardware of anoptimization device of a second embodiment.

The optimization device 20 of the second embodiment is, for example, acomputer, and includes a CPU 21, a RAM 22, an HDD 23, an image signalprocessing unit 24, an input signal processing unit 25, a medium reader26, and a communication interface 27. The above units are connected to abus.

The CPU 21 is a processor including an arithmetic circuit that executesprogram instructions. The CPU 21 loads at least part of programs anddata stored in the HDD 23 into the RAM 22 to execute programs. Note thatthe CPU 21 may include a plurality of processor cores, the optimizationdevice 20 may include a plurality of processors, and processes to bedescribed below may be performed in parallel using the plurality ofprocessors or processor cores. Furthermore, a set of a plurality ofprocessors (multiprocessors) may be called a “processor”.

The RAM 22 is a volatile semiconductor memory that temporarily storesprograms executed by the CPU 21 and data used by the CPU 21 forarithmetic operations. Note that the optimization device 20 may includeany type of memory other than RAM, or may include a plurality ofmemories.

The HDD 23 is a non-volatile storage device that stores programs ofsoftware such as an operating system (OS), middleware, and applicationsoftware, and data. The programs include, for example, a program forcontrolling the optimization device 20. Note that the optimizationdevice 20 may include another type of storage device such as a flashmemory or a solid state drive (SSD), and may include a plurality ofnonvolatile storage devices.

The image signal processing unit 24 outputs an image to a display 24 aconnected to the optimization device 20 according to a command from theCPU 21. As the display 24 a, a cathode-ray tube (CRT) display, a liquidcrystal display (LCD), a plasma display panel (PDP), an organicelectro-luminescence (OEL), or the like display may be used.

The input signal processing unit 25 obtains an input signal from aninput device 25 a connected to the optimization device 20, and outputsthe signal to the CPU 21. As the input device 25 a, a pointing devicesuch as a mouse, a touch panel, a touch pad, and a trackball, akeyboard, a remote controller, and a button switch and the like may beused. Furthermore, a plurality of types of input devices may beconnected to the optimization device 20.

The medium reader 26 is a reading device that reads a program or datarecorded on a recording medium 26 a. As the recording medium 26 a, forexample, a magnetic disk, an optical disk, a magneto-optical disk (MO),a semiconductor memory, or the like may be used. Examples of themagnetic disk include a flexible disk (FD) and an HDD. Examples of theoptical disk include a compact disc (CD) and a digital versatile disc(DVD).

The medium reader 26 copies, for example, a program or data read fromthe recording medium 26 a to another recording medium such as the RAM 22or the HDD 23. The read program is executed by the CPU 21, for example.Note that the recording medium 26 a may be a portable recording medium,and may be used for distribution of a program or data. Furthermore, therecording medium 26 a and the HDD 23 may be also referred to as acomputer-readable recording media.

The communication interface 27 is an interface that is connected to thenetwork 27 a and communicates with another information processing devicevia the network 27 a. The communication interface 27 may be a wiredcommunication interface connected by a cable to a communication devicesuch as a switch, or may be a wireless communication interface connectedto a base station via a wireless link.

Next, functions and processing procedures of the optimization device 20will be described.

FIG. 4 is a block diagram illustrating a function example of theoptimization device of the second embodiment.

The optimization device 20 has a storage unit 30 and a processing unit31. The processing unit 31 has a control unit 31 a, a setting readingunit 31 b, a problem setting unit 31 c, a Hamiltonian calculation unit31 d, an energy difference calculation unit 31 e, a random numbergeneration unit 31 f, a transition probability calculation unit 31 g, anMCMC calculation execution unit 31 h, an SA execution unit 31 i, and areplica exchange execution unit 31 j. Moreover, the processing unit 31has an energy update unit 31 k, a result output unit 31 l, a λ annealingunit 31 m, and a parameter optimization unit 31 n.

Note that the storage unit 30 may be implemented using, for example, astorage area secured in the HDD 23. The processing unit 31 may beimplemented using, for example, a program module executed by the CPU 21.

The storage unit 30 stores energy information, spin information, replicainformation, temperature information, problem setting information, andHamiltonian information.

The energy information includes an initial value of calculated energyand N_(rank) pieces of energy calculated so far from the minimum valuein ascending order. Furthermore, the energy information may includeN_(rank) combinations of values of respective state variablescorresponding to the N_(rank) pieces of energy. The spin informationincludes values of the respective state variables. The replicainformation is information used for executing the replica exchangemethod, and includes the number of replicas and the like. Thetemperature information includes a value of a temperature parameter(hereinafter, simply referred to as a temperature) or an inversetemperature. The problem setting information Includes, for example, theoptimization method to be used (the SA method, the replica exchangemethod, or the like), frequencies of replica exchange and annealing,sampling frequency, temperature change schedule, information oftransition probability distribution to be used, and the number of MCMCcalculations that is a calculation end condition. The Hamiltonianinformation includes, for example, a weighting factor (W_(ij)), a biascoefficient (b_(i)), and a constant (C) of the energy functionrepresented in equation (1).

The control unit 31 a controls each unit of the processing unit 31 toperform a minimum value solution process.

The setting reading unit 31 b reads the above-described various types ofinformation from the storage unit 30 in a format that can be understoodby the control unit 31 a.

The problem setting unit 31 c calculates an initial value of a statevariable and an initial value of a minimum value of energy.

The Hamiltonian calculation unit 31 d calculates energy.

The energy difference calculation unit 31 e calculates an energydifference due to inversion of a value of a state variable.

The random number generation unit 31 f generates a random number forrandomly selecting a state variable to be inverted. As the random numbergeneration unit 31 f, any type may be used, but it is desirable to use apseudo random number generator such as a Mersenne Twister having aperiod sufficiently larger than the number of MCMC calculations.

The transition probability calculation unit 31 g calculates a value of atransition probability distribution based on a selected equation of thetransition probability distribution, a temperature or an inversetemperature, and a calculated energy difference.

The MCMC calculation execution unit 31 h performs the MCMC calculationof determining whether or not to update the value of the selected statevariable according to a calculated value of the transition probabilitydistribution.

The SA execution unit 31 i lowers the temperature based on thetemperature change schedule, in order to execute the SA method.

The replica exchange execution unit 31 j exchanges temperatures betweenreplicas based on an exchange probability as described later, in orderto execute the replica exchange method.

The energy update unit 31 k updates the energy information stored in thestorage unit 30 based on the energy calculated by the Hamiltoniancalculation unit 31 d.

The result output unit 31 l outputs the energy information as acalculation result when the current number of MCMC calculations reachesa predetermined number (hereinafter, referred to as N_(E)) that is thecalculation end condition. Note that the result output unit 31 l mayoutput energy calculated at this moment and the value of each statevariable corresponding to the energy every number of calculations thatis smaller than N_(E) (hereinafter referred to as N_(I)). In that case,the optimization device 20 functions as a sampling device, and N_(I) isthe sampling frequency.

The λ annealing unit 31 m executes a λ annealing method (also referredto as a parameter annealing method or a two-state transition method) asdescribed later, which is an annealing method different from the SAmethod.

The parameter optimization unit 31 n performs a process of optimizingvarious parameters by a method as described later.

FIG. 5 is a flowchart illustrating a process flow of an example of amethod for controlling the optimization device of the second embodiment.

When the process starts, first, the setting reading unit 31 b reads theabove-described various types of information from the storage unit 30 ina format that can be understood by the control unit 31 a (step S10).Examples of the information to be read will be described later.Thereafter, the problem setting unit 31 c performs an initializationprocess including initialization of the state variables and calculationof an initial value of energy (step S11). Then, for example, the controlunit 31 a determines whether or not problem setting information readfrom the storage unit 30 includes information instructing to perform aparameter optimization process (step S12). When the control unit 31 adetermines that the information instructing to perform the parameteroptimization process is included, a process of step S24 is performed.

When the control unit 31 a determines that it is not specified toperform the parameter optimization process, the control unit 31 ainitializes N_(C) that indicates the current number of MCMC calculations(hereinafter, also referred to as the number of steps) to 1 (step S13).After the process of step S13, the MCMC calculation execution unit 31 hperforms an M-H calculation process by the MCMC calculation (step S14).An example of the M-H calculation process will be described later.Thereafter, the energy update unit 31 k performs an energy updateprocess of updating the energy information stored in the storage unit 30based on energy calculated by the Hamiltonian calculation unit 31 daccording to a result of the M-H calculation process (step S15).

After the process of step S15, the control unit 31 a determines whetheror not the problem setting information read from the storage unit 30includes information instructing to perform annealing (SA method or λannealing method) (step S16). When the control unit 31 a determines thatthe information instructing to perform annealing is included, anannealing process by the SA execution unit 31 i or the λ annealing unit31 m is performed (step S17). An example of the annealing process willbe described later. After the annealing process, or when the controlunit 31 a determines that the problem setting information does notinclude the information instructing to perform the annealing, a processof step S18 is performed.

In the process of step S18, the control unit 31 a determines whether ornot the problem setting information read from the storage unit 30includes information instructing to perform the replica exchange method(step S18). When the control unit 31 a determines that the informationinstructing to perform the replica exchange method is included, thereplica exchange execution unit 31 j performs a replica exchange process(step S19). An example of the replica exchange process will be describedlater. After the replica exchange process, or when the control unit 31 adetermines that the problem setting information does not include theinformation instructing to perform the replica exchange method, aprocess of step S20 is performed. Note that since the annealing processby the SA method and the replica exchange process are mutually exclusiveprocesses, either one may be selected. On the other hand, the annealingprocess, by the λ annealing method and the replica exchange process maybe simultaneously selected.

In the process of step S20 (sample output process), every time N_(I)times of calculation processes are performed, the result output unit 31l outputs energy calculated at that time and the value of each statevariable corresponding to the energy. An example of the sample outputprocess will be described later.

Next, the control unit 31 a determines whether or not N_(C)=N_(E) istrue (step S21). When it is determined that N_(C)=N_(E) is not true, thecontrol unit 31 a adds +1 to N_(C) (step S22). Thereafter, the processfrom step S14 is repeated. When the control unit 31 a determines thatN_(C)=N_(E) is true, the result output unit 31 l outputs energyinformation as a calculation result (step S23). Thereafter, theprocessing of the optimization device 20 ends. The output energyinformation includes, for example, N_(rank) pieces of energy in whichthe pieces of energy calculated so far are arranged in ascending orderfrom the minimum value. Furthermore, the energy information includesN_(rank) combinations of the values of the respective state variablescorresponding to the N_(rank) pieces of energy.

In the process of step S24, the parameter optimization unit 31 nperforms a process of optimizing various parameters by a method asdescribed later, and displays a result thereof. Thereafter, theprocessing of the optimization device 20 ends.

Note that the process flow illustrated in FIG. 5 is an example, and theorder of the processes may be appropriately changed.

Example of Information Reading Process

FIG. 6 is a flowchart illustrating a process flow of an example of aninformation reading process.

The setting reading unit 31 b reads the number of calculations (N_(E))that is the calculation end condition and the sampling frequency (N_(I))from the storage unit 30 (steps S30 and S31). Moreover, the settingreading unit 31 b reads the temperature (for example, T (temperature(absolute temperature)) that determines β(=1/T) in the above-describedequations (14) to (16)) from the storage unit 30 (step S32).Furthermore, the setting reading unit 31 b reads the problem size (thenumber of state variables (N)) and the Hamiltonian information (theweighting factor (W_(ij)), the bias coefficient (b_(i)), and theconstant (C) of the energy function represented in equation (1)) fromthe storage unit 30 (steps S33 and S34).

Furthermore, the setting reading unit 31 b reads a spin initializationmethod (how to determine initial values of state variables) andtransition probability distribution information from the storage unit 30(steps S35 and S36). The transition probability distribution informationincludes information indicating which transition probabilitydistribution is used for the MCMC calculation (distribution name and thelike) and parameters of the transition probability distribution to beused (for example, m₁ to m₃ in equations (14) to (16)).

Moreover, the setting reading unit 31 b reads the optimization method(SA method, replica exchange method, or the like) and annealinginformation (for example, temperature change schedule and annealingfrequency) to be used from the storage unit 30 (steps S37 and S38).Furthermore, the setting reading unit 31 b reads, from the storage unit30, the number of replicas, temperature information for replica exchange(a temperature sequence to be set to each replica), and frequencies ofannealing and replica exchange (steps S39, S40, and S41), and ends theinformation reading process.

Note that the process flow illustrated in FIG. 6 is an example, and theorder of the processes may be appropriately changed.

Example of Initialization Process

In the Ising model, initialization of spins (state variables) isimportant. In an example of the initialization process described below,the problem setting unit 31 c performs one of four types of spininitialization methods (“External”, “Random”, “One”, and “Zero”).

FIG. 7 is a flowchart illustrating a process flow of an example of theinitialization process.

For example, first, the problem setting unit 31 c determines whether ornot the spin initialization method is “External” (step S50). When it isdetermined that the spin initialization method is “External”, theproblem setting unit 31 c sets a state variable obtained from outsidethe optimization device 20 as an initial value of the spin information(step S51).

When it is determined that the spin initialization method is not“External”, the problem setting unit 31 c determines whether or not thespin initialization method is “Random” (step S52). When it is determinedthat the spin initialization method is “Random”, the problem settingunit 31 c randomly sets 0 or 1 to each of state variables to use it asan initial value of the spin information (step S53). For example, theproblem setting unit 31 c sets all the state variables to 0 with aprobability of 50%, and sets 1 to state variables that have not been setto 0.

When it is determined that the spin initialization method is not“Random”, the problem setting unit 31 c determines whether or not thespin initialization method is “One” (step S54). When it is determinedthat the spin initialization method is “One”, the problem setting unit31 c sets all the state variables to 1 and use them as initial values ofthe spin information (step S55).

When it is determined that the spin initialization method is not “One”,the problem setting unit 31 c determines that the spin initializationmethod is “Zero”, and sets all the state variables to 0 to take them asinitial values of the spin information to (step S56).

After the processes of steps S51, S53, S55, and S56, the problem settingunit 31 c calculates an initial value of energy (E) (step S57). Theproblem setting unit 31 c calculates an initial value of energy based onthe Hamiltonian information and the initial values of the spininformation by using equation (1).

Moreover, by setting E_(min)=E, the problem setting unit 31 c sets thecalculated initial value of energy as the minimum value of the energy(E_(min)) (step S58), and ends the initialization process.

Note that the process flow illustrated in FIG. 7 is an example, and theorder of the processes may be appropriately changed.

Example of M-H Calculation Process

FIG. 8 is a flowchart illustrating a process flow of an example of theM-H calculation process.

When the identification numbers of the N state variables are 1 to N, therandom number generation unit 31 f randomly generates integer values of1 to N to determine a state variable as an inversion candidate (stepS60).

Then, the energy difference calculation unit 31 e calculates an energydifference (ΔE) when inverting the value of the state variable as theinversion candidate (step S61). When the identification number of thestate variable as the inversion candidate is k, the energy difference isΔE_(k)=−(1−2x_(k))h_(k). h_(k) is a local field represented by equation(2).

The MCMC calculation execution unit 31 h determines whether or not theenergy difference calculated by the energy difference calculation unit31 e is smaller than 0 (step S62). When the MCMC calculation executionunit 31 h determines that the energy difference is equal to or greaterthan 0, the MCMC calculation execution unit 31 h (or the random numbergeneration unit 31 f) generates a uniform random number R in a section[0, 1] (step S63). Then, the MCMC calculation execution unit 31 hdetermines whether or not the transition probability distribution(f(ΔE)) calculated by the transition probability calculation unit 31 gis smaller than the uniform random number R (step S64).

When it is determined that f(ΔE)<R is true, or when it is determinedthat ΔE<0 is true in the process of step S62, the MCMC calculationexecution unit 31 h inverts the value of the state variable of theinversion candidate (step S65). Thus, the spin information stored in thestorage unit 30 is updated. Noted that, in view of the application todeveloping of a concrete system, the above relationship between thetransition probability distribution f(ΔE) and the random number R may bereversed. For example, the relationship to be evaluated in the processof step S64 may be adjusted such that if f(ΔE)>R is true then the MCMCcalculation execution unit 31 h inverts the value of the state variableof the inversion candidate (step S65).

Furthermore, the Hamiltonian calculation unit 31 d calculates (updates)the energy using ΔE calculated in the process of step S61 (step S66).

When the MCMC calculation execution unit 31 h determines that f(ΔE)≥R inthe process of step S64, or after the process of step 66, the MCMCcalculation execution unit 31 h ends one M-H calculation process. Asmentioned above, in view of the application to developing of a concretesystem, the MCMC calculation execution unit 31 h may be configured toends one M-H calculation process when f(ΔE)≤R is true in the process ofstep S64.

Note that the process flow illustrated in FIG. 8 is an example. Forexample, the order of the processes of steps S65 and S66 may beexchanged.

Example of Energy Update Process

FIG. 9 is a flowchart illustrating a process flow of an example of theenergy update process.

The energy update unit 31 k initializes a rank indicating how small theenergy (E) calculated by the Hamiltonian calculation unit 31 d is toN_(rank)+1 among N_(rank) pieces of energy stored in the storage unit 30as the energy information (step S70).

Then, the energy update unit 31 k sets j indicating the rank of theN_(rank) pieces of energy to 1 (step S71), and then determines whetherE<E_(min)[j] is true or not (step S72). E_(min)[j] represents the j-thsmallest energy among the N_(rank) pieces of energy.

When it is determined that E<E_(min)[j] is true, the energy update unit31 k sets the rank=j (step S73). For example, when the energy calculatedthis time falls within the N_(rank) pieces of energy (within theranking), the rank is held as rank=j in, for example, the RAM 22.

When it is determined that E≥E_(min)[j] is true, the energy update unit31 k determines whether or not j=N_(rank) is true (step S74) and, whenit is determined that j=N_(rank) is not true, sets j=j+1 (step S75) andrepeats the process from step S72.

When it is determined that j=N_(rank) is true without any change of therank=N_(rank)+1 (energy is out of the ranking), the energy update unit31 k ends the energy update process.

After the process of step S73, the energy update unit 31 k determineswhether or not the rank=N_(rank) is true (step S76). When it isdetermined that the rank=N_(rank) is not true, the energy update unit 31k sets j=rank+1 (step S77).

Then, the energy update unit 31 k sets E_(min)[j]=E_(min)[j−1] (stepS78), and sets {x_(j)}={x_(j−1)} (step S79). {x_(j)} indicates the valueof each state variable corresponding to the j-th smallest energy.

Thereafter, the energy update unit 31 k determines whether or notj=N_(rank) is true (step S80) and, when it is determined that j=N_(rank)is not true, sets j=j+1 (step S81) and repeats the process from stepS78.

When it is determined that j=N_(rank) is true or determined that therank=N_(rank) is true in the process of step S76, the energy update unit31 k sets E_(min)[rank]=E (step S82), and {x_(rank)}={x} (step S83).Thus, the energy information stored in the storage unit 30 is updated.After the process of step S83, the energy update process ends.

Note that the process flow illustrated in FIG. 9 is an example. Theorder of the processes may be appropriately changed.

Example of Annealing Process

FIG. 10 is a flowchart illustrating a process flow of an example of anannealing process.

The control unit 31 a determines whether or not the problem settinginformation read from the storage unit 30 includes informationinstructing to perform the SA method (step S90). When the control unit31 a has determined that the problem setting information includesinformation instructing to perform the SA method, the SA execution unit31 i performs an SA process (step S91). When it is determined that theproblem setting information does not include the information instructingto perform the SA method, the control unit 31 a determines whether ornot the problem setting information includes information instructing toperform the λ annealing method (step S92). When the control unit 31 adetermines that the problem setting information includes the informationinstructing to perform the λ annealing method, the λ annealing unit 31 mperforms the λ annealing process (step S93).

After the process of steps S91 and S93, or when it is determined in theprocessing in step S92 that the problem setting information does notinclude the information instructing to perform the λ annealing method,the annealing process ends.

Example of SA Process

FIG. 11 is a flowchart illustrating a process flow of an example of theSA process.

The SA execution unit 31 i determines whether or not MOD(N_(C), N_(A))=0is true (step S100). N_(C) is the current number of calculations (numberof steps), and N_(A) is the number of steps indicating the frequency ofannealing. In the process of step S100, it is determined whether or notN_(C) is a multiple of N_(A). When it is determined that MOD(N_(C),N_(A))=0 is not true, the SA execution unit 31 i ends the SA process.

When it is determined that MOD(N_(C), N_(A))=0 is true, the SA executionunit 31 i determines a temperature change schedule (step S101).

In the example of FIG. 11 , four examples of the temperature changeschedule are illustrated. “inv” is a temperature change schedule toreduce the temperature (T) in inverse proportion to N_(C). “invroot” isa temperature change schedule to reduce the value of T in inverseproportion to the square root of N_(C). “exp” is a temperature changeschedule to reduce the value of T by an exponential schedule. “list” isa temperature change schedule to change (decrease) the value of Taccording to a list (a set of an arbitrarily given number of steps andthe value of T).

When the temperature change schedule is “inv”, the SA execution unit 31l decreases the value of T in inverse proportion to N_(C) (step S102).When the temperature change schedule is “invroot”, the SA execution unit31 i decreases the value of T in inverse proportion to the square rootof N_(C) (step S103). When the temperature change schedule is “exp”, theSA execution unit 31 i decreases T by an exponential schedule (stepS104). When the temperature change schedule is “list”, the SA executionunit 31 i decreases the value of T according to the list (step S102). Anarbitrary temperature change schedule may be set by using the list.

After the processes of steps S102 to S105, the SA process ends.

Example of λ Annealing Process

In the λ annealing method, H(λ)=(1−λ)H_(A)+λH_(B) is used as an energyfunction.

Further, H_(B) is a Hamiltonian whose answer is desired, and forexample, equation (1) is used. H_(A) is a Hamiltonian in an initialstate, a known Hamiltonian whose answer is well known. For example, inconsideration of a one-dimensional arrangement of spins, it is assumedthat interaction occurs only between adjacent spins, and all strengthsof interaction are given as a constant value. In this manner, it ispossible to create a state where an analytical solution exists in H_(A).For example, as H_(A), a Hamiltonian whose energy (H(λ)) is in thelowest state when the values of all spins (state variables) are 0 may beemployed. Of course, a Hamiltonian whose energy is in the lowest statewhen the values of all the state variables are 1 may be employed asH_(A). A manner of selecting H_(A) depends on how the initial state isdetermined.

Further, λ (annealing variable) is 0≤λ≤1, and can be defined as λ=λ(t)as a function of time. However, λ(t) satisfies a boundary condition ofλ(0)=0 at a calculation start time (t=0) and λ(τ)=1 at a calculation endtime (t=τ). Thus, the Hamiltonian as a function of time is H(0)=H_(A) atthe calculation start time, and H(τ)=H_(B) at the calculation end time.

Therefore, the calculation is started from a state in which all thestate variables are initialized at t=0, and a Hamiltonian answer thatthe calculator desires to know is obtained at the end time of thecalculation. Although this is a concept of the Ising-type quantumcomputer, the optimization device 20 does not deal with the quantumtheory and uses the λ annealing method as a method of finding theminimum value of energy within the range of the classical Hamiltonian.

FIG. 12 is a flowchart illustrating a process flow of an example of theλ annealing process.

The λ annealing unit 31 m determines whether or not MOD(N_(C), N_(A))=0is true (step S110). N_(C) is the current number of calculations (numberof steps), and N_(A) is the number of steps indicating the frequency ofannealing. In the process of step S110, it is determined whether or notN_(C) is a multiple of N_(A). When it is determined that MOD(N_(C),N_(A))=0 is not true, the λ annealing unit 31 m ends the λ annealingprocess.

When it is determined that MOD(N_(C), N_(A))=0 is true, the λ annealingunit 31 m determines, for example, a λ change schedule included in theread problem setting information (step S111).

In the example of FIG. 12 , four examples of the λ change schedule areillustrated. “linear” is a λ change schedule that increases λ(N_(C))with an increase of N_(C) according to λ(N_(C))=N_(C)N_(E). “quadratic”is a λ change schedule that increases λ(N_(C)) with an increase of N_(C)according to the square of (N_(C)/N_(E)). “sqrt” is a λ change schedulethat increases λ(N_(C)) with an increase of N_(C) according to thesquare root of (N_(C)/N_(E)). “list” is a λ change schedule thatincreases λ(N_(C)) according to a function called f(N_(C)). f(N_(C)) isarbitrary, but is given as a sequence of pairs (N_(i), λ_(i)) based onthe number of steps and the value of λ at the number of steps.

When the λ change schedule is “linear”, the λ annealing unit 31 mcalculates λ(N_(C)) by calculating λ(N_(C))=N_(C)/N_(E) (step S112).When the λ change schedule is “quadratic”, the λ annealing unit 31 mcalculates λ(N_(C)) by calculating the square of (N_(C)/N_(E)) (stepS113). When the λ change schedule is “sqrt”, the λ annealing unit 31 mobtains λ(N_(C)) by calculating the square root of (N_(C)/N_(E)) (stepS114). When the λ change schedule is “list”, the λ annealing unit 31 mcalculates λ(N_(C)) according to the function f(N_(C)) described above(step S115).

After the processes of steps S112 to S115, the λ annealing unit 31 mreads from the storage unit 30 the value of each state variable thatgives the minimum value of H(λ) (for example, calculated by theHamiltonian calculation unit 31 d) for λ before the change (step S116).The read value of each state variable is used as an initial value whenthe M-H calculation process is performed again after the change of λ.After the process of step S116, the λ annealing processing ends.

Note that the λ annealing process can be performed simultaneously withthe replica exchange process described below.

Example of Replica Exchange Process

FIG. 13 is a flowchart illustrating a process flow of an example of thereplica exchange process.

The replica exchange execution unit 31 j determines whether or notMOD(N_(C), N_(R))=0 is true (step S120). N_(C) is the current number ofcalculations (the number of steps), and N_(R) is the number of stepsindicating the frequency of replica exchange. In the process of stepS120, it is determined whether or not N_(C) is a multiple of N_(R).

When it is determined that MOD(N_(C), N_(R))=0 is true, the replicaexchange execution unit 31 j determines whether or not MOD(N_(C), 2)=0is true (step S121). In the process of step S121, it is determinedwhether or not N_(C) is an even number.

When it is determined that MOD(N_(C), 2)=0 is true, the replica exchangeexecution unit 31 j selects all pairs of an even-numbered replica inorder of temperatures and a replica having one larger temperature (stepS122). When it is determined that MOD(N_(C), 2)=0 is not true, thereplica exchange execution unit 31 j selects all pairs of anodd-numbered replica in order of temperatures and a replica having onelarger temperature (step S123). By determining a replica pair as areplica exchange candidate in this manner, the efficiency of replicaexchange is increased.

After the processes of steps S122 and S123, the replica exchangeexecution unit 31 j determines the replica number of a pair for whichreplica exchange is performed (step S124).

FIG. 14 is a diagram illustrating an example of a method for determininga replica number of a pair for which replica exchange is performed.

In FIG. 14 , an upper part in parentheses indicates a replica number,and a lower part indicates a temperature set to the replica of thereplica number. However, T₁<T₂<T₃. When temperatures are exchangedbetween replicas to be exchanged in replica exchange, the correspondencebetween the replica numbers and the temperatures changes. When thereplica exchange is performed between adjacent replicas to whichtemperatures are set, in the process of step S124, replica numberscorresponding to adjacent temperatures (T_(i) and T_(i+1)) are detectedin order to detect such a replica.

In a leftmost side example in FIG. 14 , replica numbers=1 and 2corresponding to adjacent temperatures (T₁, T₂) are detected. Whentemperatures are exchanged between the replicas having the replicanumbers of 1 and 2, as illustrated in a middle example of FIG. 14 ,there are two pairs of 1 and 2 and 1 and 3 of replica numberscorresponding to adjacent temperatures. Since the same temperatures havejust been exchanged between the replica numbers=1 and 2, the replicanumbers=1 and 3 are determined as the replica numbers of a pair forwhich the replica exchange is performed.

Note that although such a process is not needed when the values of statevariables are exchanged between replicas, when the degree of freedom ofthe state variable increases, the amount of information generated bytransmission and reception at the time of exchange increases, and it isdesirable to exchange temperatures as described above.

Thereafter, the replica exchange execution unit 31 j sets id (=0 toN_(rep)−1 (N_(rep) is the number of pairs selected in steps S122 andS123)) of the pair for which the replica exchange is to be performed to0 (step S125). Then, the replica exchange execution unit 31 j determineswhether or not to execute the replica exchange for the pair of id=j(step S126).

The exchange probability in the replica exchange method can be expressedby the following equation (18) when the transition probabilitydistribution to be used satisfies the principle of detailed balancesimilarly to the Boltzmann distribution.

$\begin{matrix}{P_{{A->B}\;} = {{\min\left\{ {1,\frac{P_{B}(t)}{P_{A}(t)}} \right\}} = {\min\left\{ {1,\frac{{\pi\left( {\beta_{i},\left\{ x_{j} \right\}} \right)}{\pi\left( {\beta_{j},\left\{ x_{i} \right\}} \right)}}{{\pi\left( {\beta_{i},\left\{ x_{i} \right\}} \right)}{\pi\left( {\beta_{j},\left\{ x_{j} \right\rbrack} \right)}}} \right\}}}} & (18)\end{matrix}$

In equation (18), P_(A)(t) is a probability that a state A before thereplica exchange is realized. In the state A, the state variable in thereplica of the replica number=i to which β_(i) is set follows aprobability distribution π(β_(i), {x_(i)}), and the state variable inthe replica of the replica number=j to which β_(j) is set follows aprobability distribution π(β_(j), {x_(j)}). β_(i) and β_(j) are inversetemperatures. In equation (18), P_(B)(t) is a probability that a state Bafter the replica exchange is realized. In the state B, the statevariable in the replica of the replica number=i to which β_(i) is setfollows a probability distribution π(β_(i), {x_(j)}), and the statevariable in the replica of the replica number=j to which β_(j) is setfollows a probability distribution π(β_(j), {x_(i)}). Although equation(18) represents a case where the state variables {x_(i)} and {x_(j)} areexchanged between replicas with replica numbers=i and j, the sameexchange probability P_(A→B) is used when β_(i) and β_(j) are exchanged.

On the other hand, when the transition probability distribution to beused does not satisfy the detailed balance, the exchange probabilityP_(A→B) is represented by the following equation (19).

$\begin{matrix}{P_{A->B} = {{\min\left\{ {1,\frac{P_{B}(t)}{P_{A}(t)}} \right\}} = {\min\left\{ {1,{f\left( {\beta_{i}\Delta\; E} \right)}} \right\}}}} & (19)\end{matrix}$

In equation (19), ΔE is E_(j)−E_(i). For example, ΔE is the differencebetween energy in the replica with the replica number=j to which β_(j)is set and energy in the replica with the replica number=i to whichβ_(i) is set.

The reason for using equation (19) will be described. When thetransition probability is defined by the above-described equation (9),as described above, a stochastic matrix having an eigenvalue of 1 isobtained, and a stationary Markov chain is formed. For example, thestate converges to a stationary state. State density at the time ofconvergence to this stationary state becomes state density correspondingto the transition probability. Now, when all replicas are exchanged atthe exchange probability of equation (19) at the time of replicaexchange, the stationary state also exists in a system with all thereplicas. When an extended ensemble method such as a replica exchangemethod is used, it is premised that a Markov chain forms a stationarystate, and therefore, an exchange probability for obtaining a stationarystate like equation (19) is used.

As f(β_(i)ΔE) in equation (19), a transition probability distribution asillustrated in equations (14) to (16) is used.

In the process of step S126, the replica exchange execution unit 31 jdetermines whether or not to execute the replica exchange on a pair ofid=j according to the exchange probability of equation (19). Forexample, based on a comparison result between f(β_(i)ΔE) in equation(19) and a uniform random number R satisfying 0≤R≤1, the replicaexchange execution unit 31 j determines to execute the replica exchangeif f(β_(i)ΔE)<R is true. If f(β_(i)ΔE)<R is not true, the replicaexchange execution unit 31 j determines not to execute the replicaexchange. Noted that, in view of the application to developing of aconcrete system, the above relationship between the f(β_(i)ΔE) and therandom number R may be reversed. For example, the relationship may beadjusted such that if f(β_(i)ΔE)>R is true then the replica exchangeexecution unit 31 j determines to execute the replica exchange. In thiscase, the replica exchange execution unit 31 j determines not to executethe replica exchange when f(β_(i)ΔE)>R is not true.

When it is determined to execute the replica exchange, the replicaexchange execution unit 31 j executes the replica exchange by exchangingtemperatures between the replicas included in the pair of id=j (stepS127).

When it is determined not to execute the replica exchange or after theprocess of step S127, the replica exchange execution unit 31 jdetermines whether or not j=N_(rep)−1 is true (step S128). When it isdetermined that j=N_(rep)−1 is true, the replica exchange execution unit31 j ends the replica exchange process, or when it is determined thatj=N_(rep)−1 is not true, j=j+1 is set (step S129), and the process fromS126 is repeated.

Example of Sample Output Process

FIG. 15 is a flowchart illustrating a process flow of an example of asample output process. FIG. 15 illustrates an example of the sampleoutput process when the replica exchange process is performed.

The result output unit 31 l determines whether or not MOD(N_(C),N_(I))=0 is true (step S130). N_(C) is the current number ofcalculations (the number of steps), and N_(I) is the number of stepsindicating the sampling frequency. When the result output unit 31 ldetermines that MOD(N_(C), N_(I))=0 is not true, the sample outputprocess is ended.

When it is determined that MOD(N_(C), N_(I))=0 is true, the resultoutput unit 31 l sets the replica number (r) to 1 (step S131). After theprocess of step S131, the result output unit 31 l outputs the currentenergy (E_(r)) of the replica having the replica number=r (step S132),and outputs the current value of each state variable ({x_(r)}) of thereplica (step S133). Note that when the λ annealing method is performed,the result output unit 31 l outputs H(λ) for each λ value as energy.

Moreover, the result output unit 31 l outputs the N_(C), the temperature(T_(r)) set to the replica having the replica number=r, and the replicanumber (r) (step S134).

Thereafter, the result output unit 31 l determines whether or notr=N_(r) is true (step S135). N_(r) is the number of replicas. The resultoutput unit 31 l ends the sampling output process when it is determinedthat r=N_(r) is true, or sets r=r+1 (step S136) and repeats the processfrom step S132 when it is determined that r=N_(r) is not true.

(Parameter Optimization Process)

When heuristically finding a minimum value of energy, the calculatorwants to know an optimum calculation result by a minimum number ofcalculations or a minimum time. However, depending on the transitionprobability distribution used, the solution may be trapped at a localminimum and may not escape easily. In this case, as a result, samplingmay not be performed widely in an energy space (sampling efficiencydeteriorates), and accuracy and speed of obtaining the minimum value ora next best solution deteriorates. For this reason, when calculating theminimum value solution problem, selection of the transition probabilitydistribution is important.

FIG. 16 is a diagram illustrating an example of comparing samplingresults when two types of transition probability distributions are used.On the left of FIG. 16 , there is illustrated a sampling result when thetransition probability distribution represented by equation (16)(hereinafter also referred to as a tempered Gaussian-type transitionprobability distribution) is used, in which m₃=¼ is set. On the right ofFIG. 16 , there is illustrated a sampling result when the transitionprobability distribution represented by equation (14) (hereinafter alsoreferred to as a power law-type transition probability distribution) isused, in which m₁≈1 is set. However, in both the distributions, thetemperature (absolute temperature) is set to 0.1.

The horizontal axis represents Monte Carlo Steps (corresponding to thenumber of steps described above), and the vertical axis represents CostFunction (corresponding to the above-described energy).

In the example of FIG. 16 , a region where variance of sampling resultsis large (sampling range is wide) and energy is lower can be searchedwhen the tempered Gaussian-type transition probability distribution isused.

However, changing parameters such as temperature can change theconclusion.

FIG. 17 is a diagram illustrating an example of comparing samplingresults when two types of transition probability distributions withtemperatures being changed are used. FIG. 17 illustrates samplingresults when the tempered Gaussian-type transition probabilitydistribution and the power law-type transition probability distributionare used under the same conditions as in FIG. 16 except that thetemperature is changed to 1.0.

When the temperature is set to 1.0, as illustrated in FIG. 17 , a regionwhere energy is lower can be searched when the power law-type transitionprobability distribution is used. On the other hand, variance of thesampling result is large when the tempered Gaussian-type transitionprobability distribution is used.

As described above, when calculating the minimum value solution problem,it is important to select an optimal transition probability distributionand to estimate optimal parameters.

On the other hand, procedures for selecting an optimal transitionprobability distribution and estimating optimal parameters arecomplicated. This is because many calculations are performed whilechanging the used transition probability distribution and parametersbefore performing a main calculation of the minimum value solutionproblem.

Accordingly, the parameter optimization unit 31 n has a function toautomatically estimate an optimal parameter included in the transitionprobability distribution as follows. Note that it is desirable that theautomatic estimation function satisfies the following characteristics.For example, there are four characteristics such that the estimation canbe performed by a brief preliminary calculation, the average value ofenergy at the time of sampling is small, variance of energy at the timeof sampling is large, and the minimum value of energy is smaller.

FIG. 18 is a flowchart illustrating a process flow of an example of aparameter optimization process.

The parameter optimization unit 31 n reads the function name of atransition probability distribution to be optimized from the storageunit 30 (step S140). For example, the function name of the transitionprobability distribution represented by equation (14) is specified as“Power law”, the function name of the transition probabilitydistribution represented by equation (15) is specified as “Hyperbolic”,and the function name of the transition probability distributionrepresented by equation (16) is specified as “tempered Gaussian”.Furthermore, when the transition probability distribution specified bythe user is read, a function name such as “User-Defined” is specified.When reading the “User-Defined”, the parameter optimization unit 31 nalso reads a combination of (ΔE_(i), f(ΔE_(i))) and smoothly connecteach point by each combination so as to satisfy 0≤f(ΔE_(i))≤1. At thistime, the parameter optimization unit 31 n can use, for example, a cubicspline. Note that the parameter optimization unit 31 n sets f(ΔE_(i))=0when a maximum value of ΔE_(i) is given and the read ΔE_(i) is largerthan the maximum value.

Then, the parameter optimization unit 31 n reads the name of a parameterto be optimized from the storage unit 30 (step S141). A parameter commonto all the functions of the transition probability distributions istemperature (represented by an inverse temperature (β=1/T) in equations(14) to (16)), but in the cases of equations (14) to (16), m₁ to m₃ canalso be selected as parameters to be optimized.

Moreover, the parameter optimization unit 31 n reads the range foroptimizing the parameter from the storage unit 30 (step S142). Forexample, when the parameter is temperature, minimum and maximumtemperatures are read. Note that when a range different from the rangestored in the storage unit is specified, for example, the range to bespecified is input from the input device 25 a. For example, when theparameter is temperature, the range is 0 or more because negativeabsolute temperature is not defined in the optimization device 20 of thepresent embodiment.

Next, the parameter optimization unit 31 n reads the number of times ofchanging the parameter (number of preliminary calculations) N_(p) fromthe storage unit 30 (step S143). Then, the parameter optimization unit31 n calculates a parameter step width ΔP (step S144). Assuming that themaximum value of the parameter is P_(max) and the minimum value of theparameter is P_(min), the parameter optimization unit 31 n obtains ΔP bycalculating, for example, ΔP=(P_(max)−P_(min))/(N_(p)−1). Note that theparameter optimization unit 31 n may read a list indicating ΔP obtainedby unequally dividing P_(min) and P_(max).

Thereafter, the parameter optimization unit 31 n reads the number ofcalculations N_(dump) to be cut off from the storage unit 30 (stepS145). N_(dump) is the number of calculations that determines a periodcalled “burn-in” in the field of numerical statistics, and means thenumber of steps until the Markov chain converges to a stationary state.The number of burn-in steps differs depending on the problem to becalculated.

Note that in the description so far, the parameter optimization unit 31n reads various types of information from the storage unit 30, but mayalso display reading candidates on the display 24 a illustrated in FIG.3 and read information input by an operation of the input device 25 a bythe user.

After the process of step S145, the parameter optimization unit 31 nsets i indicating the current number of preliminary calculations to 1(step S146), and calculates a parameter value (step S147). In theprocess of step S147, the parameter optimization unit 31 n calculatesthe value of the parameter (P_(i)) by calculating P_(i)=P_(min)+(i−1).

Thereafter, the parameter optimization unit 31 n sets j indicating thecurrent number of MCMC calculations to 1 (step S148), and calculatesenergy (E_(i)) and energy difference (ΔE_(i)) by the MCMC calculation(step S149).

Then, the parameter optimization unit 31 n determines whether or notj>N_(dump) is true (whether or not the number of calculations hasexceeded the burn-in period) (step 150). When it is determined thatj>N_(dump) is true, the parameter optimization unit 31 n stores E_(i),ΔE_(i), and minimum value of energy in, for example, the RAM 22illustrated in FIG. 3 (step S151).

When it is determined that j>N_(dump) is not true (determined that theMarkov chain has not reached the stationary state), or after the processof step S151, the parameter optimization unit 31 n determines whether ornot j=M is true (step S152). M is the number of times of MCMCcalculation. If the value of M is taken too large, the preliminarycalculation takes a long time. Thus, the number of times is specifiedthat is larger than the number of calculations corresponding to theburn-in period, and that is needed for calculation of a statistic amountusing data after cutting off the data until the burn-in period elapses.The amount of calculation for letting a Monte Carlo integration convergeis proportional to 1/(square root of M), and thus the number ofcalculations to an extent that allows estimation is specified. Forexample, M is about 1000 to several million times, and is read from thestorage unit 30. Thus, the calculation of an evaluation value describedbelow is performed in a shorter time than the calculation time (maincalculation time) when searching for the minimum value of the energyfunction.

When it is determined that j=M is not true, the parameter optimizationunit 31 n sets j=j+1 (step S153), and repeats the process from stepS149. When it is determined that j=M is true, the parameter optimizationunit 31 n calculates an average value and a standard deviation of thestored E_(i) and ΔE_(i) (step S154). Note that the calculated averagevalue and standard deviation are stored together with the minimum valueof energy, for example, in the RAM 22 as an evaluation value of a valueof the energy function at a value of each parameter.

Thereafter, the parameter optimization unit 31 n determines whether ornot i=N_(p) is true (step S155). When it is determined that i=N_(p) isnot true, the parameter optimization unit 31 n sets i=i+1 (step S156),and repeats the process from step S147.

When it is determined that i=N_(p) is true, the parameter optimizationunit 31 n outputs the stored evaluation value (step S157), and ends theparameter optimization process. The evaluation value is output(displayed) on the display 24 a Illustrated in FIG. 3 , for example.Note that the evaluation value may be output to the storage unit 30.

Note that the process flow illustrated in FIG. 18 is an example. Theorder of the processes may be appropriately changed.

FIG. 19 is a diagram illustrating an output example of evaluationvalues.

FIG. 19 illustrates a display example of evaluation values for thetemperature on the display 24 a. An example in which calculation isperformed by N_(p)=10 from T=1 to 10 is illustrated. The minimum valueof energy (E_(min)), average value of energy (E_(ave)), standarddeviation of energy (E_(stddev)), average value of energy difference(dE_(ave)), and standard deviation of energy difference (dE_(stddev))for each temperature are listed as evaluation values. Furthermore, inthe example of FIG. 19 , respective temperatures are ranked in ascendingorder of E_(min). When N_(p) increases, the data amount of thecalculation result increases. Thus, ranking the values of respectiveparameters as described above makes it easy to determine the values ofgood parameters and allows reducing the analysis time. Note that theranking is not limited to being performed in ascending order of E_(min),and may be performed in ascending order of E_(ave).

By performing the parameter optimization process as described above,labor of the calculator in determining parameters is reduced, andparameters that can be expected to obtain good calculation results canbe easily determined.

(Effects)

Also in the optimization device 20 of the second embodiment as describedabove, the following effects similar to those of the optimization deviceof the first embodiment may be obtained.

For example, since a transition probability distribution in which atransition probability when ΔE is (positively) large is larger than theBoltzmann distribution is applied, a solution can escape from a localsolution with high efficiency. Furthermore, since it is not a method ofadding an offset to energy as in the dynamic offset method, the Markovchain may not be broken. From the above, it is possible to efficientlysearch for a minimum value of an energy function without breaking theMarkov chain.

Moreover, the optimization device 20 employs an exchange probabilityusing the transition probability distribution as described above at thetime of the replica exchange process, and thus a replica exchangefrequency is higher than when the Boltzmann distribution is used. Thus,even when the number of replicas is small, each replica randomly walksin the temperature space. Consequently, the sampling space is expanded,sampling efficiency is improved, and a more accurate solution can beobtained.

Furthermore, in the energy update process illustrated in FIG. 9 , theoptimization device 20 updates energy information so that energycalculated so far includes N_(rank) pieces of energy in ascending orderfrom the minimum value. Therefore, even when a true minimum value of theenergy function may not be obtained, a next best minimum value(solution) can be obtained.

Calculation Example

Hereinafter, in order to illustrate the above effects, a calculationexample of the traveling salesman problem by the optimization device 20will be described.

The traveling salesman problem is a problem to find the minimum distanceof a journey such that, when N cities are defined as points P_(i) (i=1,. . . , N), a salesman visits each city exactly once and finally returnsto the first city. Here, a distance d_(ij) between points P_(i) andP_(j) represents a distance between cities. In order to convert thetraveling salesman problem into an Ising model, if a state variableindicating whether or not to visit a city a in the i-th place isn_(i, a), the Ising-type energy function (E) having N² state variablescan be expressed as the following equation (20).E=½Σ_(i,j,a) ^(N) d _(ij) n _(i,a)(n _(j,a+1) +n _(j,a−1))  (20)

However, if equation (20) remains as it is, a self-evident solutionwhere the salesman does nothing (all state variables are 0) and even anunrealistic solution where the salesman visits all cities in the i-thplace are tolerated, and thus constraint conditions represented by thefollowing equations (21) and (22) are imposed.Σ_(a=1) ^(N) n _(i,a)=1  (21)Σ_(i=1) ^(N) n _(i,a)=1  (22)

The following equation (23) taken into equation (20) using theconstraint conditions expressed by equations (21) and (22) as penaltyterms is an equation to be subjected to energy minimization.E=1/1Σ_(i,j,a) ^(N) d _(ij) n _(i,a)(n _(j,a+1) +n _(j,a−1))+k ₁Σ_(i=1)^(N)(Σ_(a=1) ^(N) n _(i,a)−1)² +k ₂Σ_(a=1) ^(N)(Σ_(f=1) ^(N) n_(i,a−)1)²  (23)

In equation (23), k₁ and k₂ are constants representing the magnitude ofa penalty. By setting the magnitude of a penalty larger than the minimumvalue of energy, a self-evident solution or an unreal solution obtainedhas large energy and is removed from solution candidates.

Note that equation (23) is calculated after being converted into, forexample, the format illustrated in equation (1). k₁ and k₂ are reflectedon the weighting factor (W_(ij)).

The traveling salesman problem as described above is employed as aproblem to be calculated for the following reasons. The first reason isthat the traveling salesman problem is not a physics problem and doesnot need to be tied to the Boltzmann distribution. Further, the secondreason is that the traveling salesman problem can be converted into anIsing model. Furthermore, the third reason is that in equation (23),strengths of interaction of second-order terms are scattered, and aninteraction between distant spins (between state variables) is alsostrong. Moreover, the fourth reason is that since the constraintcondition is included in equation (23) as a penalty term, it is ideal asan Ising model with a penalty (Ising model with an external field).

FIG. 20 is a diagram illustrating an example of stationary statecalculation results when three types of transition probabilitydistributions are used. The horizontal axis represents the number ofsteps, and the vertical axis represents energy (E). FIG. 20 illustrates,from the left, sampling results when the Boltzmann distribution, thepower law-type transition probability distribution, and the temperedGaussian-type transition probability distribution are used as thetransition probability distribution.

The traveling salesman problem used is ulysees16 of TSPLIB (TSPLIB,[online], [searched on Apr. 11, 2019], Internet <URL:https://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPIB95/>).Note that calculation conditions are: T=100.0, k₁ and k₂ in equation(23) are 150, N_(E)=100,000, N_(I)=10, m₁ in equation (14) is 1.02, andm₃ in equation (16) is 0.25.

As illustrated in FIG. 20 , when the power law-type and temperedGaussian-type transition probability distributions are used, it can beseen that convergence to a stationary state occurs by repeating theMarkov chain, similarly to when the Boltzmann distribution is used.Moreover, the search range for energy is about [1000, 4000] when theBoltzmann distribution is used, but expands to about [1000, 9000] in thepower law-type transition probability distribution. The same applieswhen the tempered Gaussian-type transition probability distribution isused.

FIG. 21 is a diagram illustrating an example of comparing searchperformance. The horizontal axis represents the number of steps, and thevertical axis represents energy (E). FIG. 21 illustrates a samplingresult when the Boltzmann distribution is used as the transitionprobability distribution (upper left), and a sampling result by a methodusing the Boltzmann distribution as the transition probabilitydistribution and adding an offset to energy (dynamic offset method)(upper right). Moreover, a sampling result when the power law-typetransition probability distribution is used as the transitionprobability distribution (lower left) and a sampling result when thetempered Gaussian-type transition probability distribution is used(lower right) are illustrated.

The traveling salesman problem used is ulysees16 similarly to the above,and calculation conditions are: T=0.25, k₁ and k₂ in equation (23) are150, N_(E)=200,000, N_(I)=10, m₁ in equation (14) is 1.02, and m₃ inequation (16) is 0.25. Note that in the calculation of the dynamicoffset method, when the same state is continuously maintained 50 times,it is considered that a solution has fallen to the local minimum, andthe offset is added.

As is clear from FIG. 21 , when the Boltzmann distribution is used asthe transition probability distribution without using the offset, once asolution falls into a local minimum, the solution does not come out ofthe local minimum. When the dynamic offset method is applied, thesolution can easily come out of the local minimum, but there are onlyfew points in an energy region that is desired to be known as thesolution (there are few good solutions). On the other hand, when thepower law-type transition probability distribution or the temperedGaussian-type transition probability distribution is used, it ispossible to escape from the local minimum and find many solutions in alower energy region.

Thus, it can be seen that when the power law-type or temperedGaussian-type transition probability distribution is used, the searchperformance improves compared to when the Boltzmann distribution is usedor when the dynamic offset method using the Boltzmann distribution isapplied.

Moreover, when the power law-type or tempered Gaussian-type transitionprobability distribution is used, an extended ensemble method such as areplica exchange method can be applied as it is because it is not amethod of adding an offset to energy and does not break the Markovchain.

Hereinafter, calculation examples when the replica exchange method isperformed using the Boltzmann distribution as the transition probabilitydistribution and when the replica exchange method is performed using thepower law-type transition probability distribution will be described.The traveling salesman problem used is ulysees16 as above, andcalculation conditions are: k₁ and k₂ in equation (23) are 150,N_(E)=200,000 (for every replica), N_(I)=10, and m₁ in equation (14) is1.02. Note that the number of replicas is eight, and a temperaturesequence set to respective replicas is T=1.0, 3.0, 5.0, 10.0, 15.0,20.0, 25.0, 30.0.

FIG. 22 is a diagram illustrating an example of calculation results oftemperature transitions in respective replicas when the Boltzmanndistribution is used. FIG. 22 illustrates a calculation result oftransition of temperature (T) in each of the eight replicas with replicanumbers=1 to 8. The horizontal axis represents the number of steps, andthe vertical axis represents T, which represents a positive real number.

As illustrated in FIG. 22 , it can be seen that when the Boltzmanndistribution is used, each replica may not be said to be randomlywalking in the temperature space with an equal probability. For example,for the replica with a replica number=5, the number of steps is fixed atT=1.0 from around 200.

FIG. 23 is a diagram illustrating an example of calculation results ofenergy in respective replicas when the Boltzmann distribution is used.FIG. 23 illustrates calculation results of energy (E) in each of theeight replicas with replica numbers=1 to 8. The horizontal axisrepresents the number of steps, and the vertical axis represents energy(E).

As Illustrated in FIG. 23 , it can be seen that when the Boltzmanndistribution is used, energy of most replicas is constrained to a localminimum and may not escape.

FIG. 24 is a diagram illustrating an example of calculation results oftemperature transitions in the respective replicas when the powerlaw-type transition probability distribution is used. FIG. 24illustrates calculation results of transitions of temperatures (T) inthe eight respective replicas with replica numbers=1 to 8. Thehorizontal axis represents the number of steps, and the vertical axisrepresents T, which represents a positive real number.

As Illustrated in FIG. 24 , it can be seen that when the power law-typetransition probability distribution is used, each temperature is setwith almost equal probability in each of the replicas, and each of thereplicas randomly walks in the temperature space.

FIG. 25 is a diagram illustrating an example of calculation results ofenergy in the respective replicas when the power law-type transitionprobability distribution is used. FIG. 25 illustrates calculationresults of energy (E) in each of the eight replicas with replicanumbers=1 to 8. The horizontal axis represents the number of steps, andthe vertical axis represents energy (E).

As illustrated in FIG. 25 , it can be seen that when the power law-typetransition probability distribution is used, energy of each replica caneasily escape when the energy falls into a specific local minimum.

FIG. 26 is a diagram illustrating an example of calculation results ofthe replica exchange method when the Boltzmann distribution and thepower law-type transition probability distribution are used.Furthermore, FIG. 27 is an enlarged view of regions where the energy issmall in the calculation results of FIG. 26 . In FIGS. 26 and 27 , thehorizontal axis represents the number of steps, and the vertical axisrepresents energy (E).

As can be seen from FIG. 26 , it can be seen that when the Boltzmanndistribution is used (left), energy is captured by a specific localminimum, but when the power law-type transition probability distributionis used (right), energy can easily escape when the energy falls into aspecific local minimum. Furthermore, an energy region where sampling canbe performed is wider when the power law-type transition probabilitydistribution is used than when the Boltzmann distribution is used.

Moreover, as can be seen from FIG. 27 , the minimum value of energyobtained is smaller when the power law-type transition probabilitydistribution is used than when the Boltzmann distribution is used.Therefore, minimum value search performance is more excellent when thepower law-type transition probability distribution is used than when theBoltzmann distribution is used. This is because when the power law-typetransition probability distribution is used, since it has relativelylarge transition probability even when an energy difference is large,the exchange probability can be earned even if temperature setting iscoarse, which is different from when the Boltzmann distribution is used.

FIG. 28 is a diagram illustrating an example of calculation results ofthe λ annealing method. The left diagram in FIG. 28 illustrates acalculation result in a case of the λ annealing method alone, and theright diagram illustrates a calculation result in a case of combiningthe λ annealing method with the replica exchange method. The horizontalaxis represents A, and the vertical axis represents energy (E) (=H(λ)).

The traveling salesman problem used is ulysees16 as above, andcalculation conditions are: k₁ and k₂ of equation (23)=150,N_(E)=200,000 (for every replica in a case of the replica exchangemethod), N_(I)=10, m₁ in equation (14) is 1.02. As for the temperature,T=1.0 is set in the case of the λ annealing method alone, and T=1.0,3.0, 5.0, 10.0, 15.0, 20.0, 25.0, 30.0 are set to the eight replicas inthe case of combining with the replica exchange method.

Annealing about λ is guaranteed to obtain aground state by sufficientlygradually increasing λ little by little in a case of an Ising-typequantum computer. However, the optimization device 20 that does nothandle the quantum theory and obtains a minimum value of energy withinthe range of the classical Hamiltonian is not guaranteed to obtain thatbecause there is no quantum effect. In the example of the left diagramof FIG. 28 , sampling efficiency is degraded as λ increases. However, bycombining the λ annealing method and the replica exchange method asillustrated in the right diagram of FIG. 28 , sampling efficiency can beimproved. Consequently, it is possible to efficiently heuristicallysolve a minimum value.

FIG. 29 is a diagram illustrating an example of comparing a calculationresult by a device that performs a parallel trial using a digitalcircuit with a calculation result by the optimization device of thesecond embodiment. The horizontal axis represents a ranking indicatingan ascending order of energy, and the vertical axis represents energy(E).

The problem to be calculated is ulysees16 as above, and k₁ and k₂ inequation (23) are 150.

A calculation result 40 is, for example, a calculation result when usinga device that performs parallel trial using a digital circuit disclosedin Non Patent Literature 1 (the SA method and the dynamic offset methodare applied). The number of trials (corresponding to N_(E) describedabove) is 200 million times.

A calculation result 41 is a calculation result when the optimizationdevice 20 of the second embodiment is used. Note that the calculationresult 41 indicates a calculation result when the annealing process andthe replica exchange process described in the process flow illustratedin FIG. 5 are not performed. N_(E)=200,000, and N_(I)=10. In theoptimization device 20, the dock frequency of the CPU 21 used is 2.60GHz, and the capacity of the RAM 22 is 12288 MB. Note that the programis implemented in C++ and is not parallelized by OpenMP or MessagePassing Interface (MPI), but calculation by one CPU core is performed.

As illustrated in FIG. 29 , when the optimization device 20 is used, asmaller minimum value is obtained and calculation accuracy is superiorto when the device that performs the parallel trial using the digitalcircuit is used. Furthermore, the calculation time is 13.5 seconds inthe former case and 1.6 seconds in the latter case, and the latter isalso superior in calculation speed.

Next, a calculation example when energy is represented by a continuousfunction will be described.

The energy function used is represented by the following equation (24)described in Non Patent Literature 3.E({x})=Σ_(i=1) ⁴(x _(i) ²−8)²+5Σ_(i=1) ⁴ x _(i) +E ₀  (24)

This energy function is a function that takes a minimum value 0 whenE₀≈57.3276. In equation (24), a defining region of x_(i) (statevariable) is an arbitrary real number. Also for the energy functiondefined by such real numbers, search for the minimum value of the energyfunction can be efficiently performed by performing the MCMC calculationusing equation (9) using the power law-type transition probabilitydistribution represented by equation (14).

FIG. 30 is a diagram illustrating an example of comparing searchperformance for an energy function that is a continuous function. Thehorizontal axis represents the number of steps, and the vertical axisrepresents energy (E) (the value of the energy function of equation(24)). FIG. 30 illustrates a sampling result when the Boltzmanndistribution is used as the transition probability distribution (left)and a sampling result when the power law transition probabilitydistribution is used (right).

The calculation conditions are: T=1.0, N_(E)=100,000, N_(t)=10, and m₁in equation (14) is 1.02.

As is clear from FIG. 30 , when the Boltzmann distribution is thetransition probability distribution, once a solution falls into a localminimum, the solution does not come out of the local minimum. On theother hand, it is clear from FIG. 30 that when the power law-typetransition probability distribution is used, it is possible to escapefrom the local minimum. Furthermore, when the power law-type transitionprobability distribution is used, as is clear from FIG. 30 , a largenumber of states such as staying in the same state a fixed number oftimes can be obtained with a small number of calculations. For example,since a large number of local minima are obtained, an efficient solutioncandidate can be obtained in a system in which a local minimum has animportant meaning besides a minimum value. Representative examples ofsuch systems include organic compounds and biomolecules.

Modification Examples

In the above description, the processing unit 31 illustrated in FIG. 4has been described as being implementable using a program moduleexecuted by the CPU 21, but each unit of the processing unit 31 may beimplemented by an electronic circuit for a specific use such as an ASICor an FPGA.

For example, the Hamiltonian calculation unit 31 d may calculateequation (1) using a product-sum operation circuit. The MCMC calculationexecution unit 31 h may also be implemented using a comparison circuitthat compares uniform random numbers with f(ΔE) as those represented inequations (14) to (16). Moreover, the energy difference calculation unit31 e may be implemented by an electronic circuit that calculates anenergy difference due to a change in value of each state variable inparallel as described in Non Patent Literature 1 or the like. Althoughthe optimization device 20 can allow a solution to escape from a localminimum with high efficiency by using a transition probabilitydistribution different from the Boltzmann distribution as describedabove, using the electronic circuit as described above as an acceleratorcan facilitate escape when the solution falls into a deep local minimum.For the same reason, a GPGPU or the like may be used as an acceleratorto execute the processing of each unit of the processing unit 31.

Note that further improvement in performance is expected by performingparallelization using OpenMP or MPI, or performing specialized tuningfor a specific CPU instruction set such as Advanced Vector eXtensions(AVX) 256 or AVX-512.

Note that in the above description, the traveling salesman problem hasbeen described as an example of a problem to be calculated, but thepresent embodiment may be applied to problems in other fields such asfinancial engineering. Furthermore, since a job scheduling problem canbe formulated using the Ising model, the present embodiment may beapplied to various schedule optimization problems (for example, aproblem of optimizing a schedule of persons concerned in a hospital orthe like). Moreover, also in the deep learning field, since a restrictedBoltzmann machine can be formulated using the Ising model, the presentembodiment may be used for optimization of the restricted Boltzmannmachine. Thus, the present embodiment may be applied to the field ofartificial intelligence.

Furthermore, other than the search for a minimum value of the Ising-typeevaluation function, the present embodiment may be applied to the searchfor a minimum value of an evaluation function in which state variablesare continuous variables, and thus the problem to be calculated is notlimited to the above-described field.

Furthermore, although a minimum value of the evaluation function issearched for in the above description, the present embodiment may beextended to a method of searching for a maximum value of the evaluationfunction by inverting the sign of the evaluation function.

Note that as described above, the above processing contents may beimplemented by causing the optimization device 20 to execute a program.

The program may be recorded on a computer-readable recording medium (forexample, the recording medium 26 a). As the recording medium, forexample, a magnetic disk, an optical disk, a magneto-optical disk, asemiconductor memory, or the like may be used. Examples of the magneticdisk include an FD and an HDD. Examples of the optical disc include aCD, a CD-R (Recordable)/RW (Rewritable), a DVD, and a DVD-R/RW. Theprogram may be recorded on a portable recording medium and distributed.In this case, the program may be copied from the portable recordingmedium to another recording medium (for example, the HDD 23) and thenexecuted.

In the foregoing, one aspect of the optimization device, the method forcontrolling the optimization device, and the program for controlling theoptimization device of the present embodiment has been described on thebasis of the embodiments; however, these are only examples and are notlimited to the above description.

All examples and conditional language provided herein are intended forthe pedagogical purposes of aiding the reader in understanding theinvention and the concepts contributed by the inventor to further theart, and are not to be construed as limitations to such specificallyrecited examples and conditions, nor does the organization of suchexamples in the specification relate to a showing of the superiority andinferiority of the invention. Although one or more embodiments of thepresent invention have been described in detail, it should be understoodthat the various changes, substitutions, and alterations could be madehereto without departing from the spirit and scope of the invention.

What is claimed is:
 1. An optimization device comprising: a memoryconfigured to store setting information indicating an optimizationmethod to be used, the memory being configured to store a value of astate variable included in an evaluation function, the evaluationfunction being obtained by converting a problem; and a processor circuitcoupled to the memory, the processor circuit being configured to select,based on the setting information stored in the memory, the optimizationmethod to be used from among a plurality of optimization methodsincluding a simulated annealing (SA) method and a replica exchangemethod, perform, in the processor circuit, the selected optimizationmethod on the evaluation function by repeating a process of updating thevalue of the state variable by the selected optimization method, theupdating of the value of state variable being performed in accordancewith a first transition probability distribution that is represented bya first function, the first function being a function configured tonormalize to one a sum of probabilities of transition from a currentstate to each of a plurality of different states, the current statebeing represented by a current value of the state variable, the firsttransition probability distribution having a feature that a transitionprobability in the first transition probability distribution is largerthan a transition probability in a Boltzmann distribution as a change inthe value of the evaluation function due to a change in the value of thestate variable is positively larger.
 2. The optimization deviceaccording to claim 1, wherein the first transition probabilitydistribution is a transition probability distribution represented by areciprocal of an m-th power (m>1) of a value that is obtained by addingone to a product of a change in a value of the evaluation function andan inverse temperature.
 3. The optimization device according to claim 1,wherein the first transition probability distribution is a transitionprobability distribution represented by a reciprocal of a (m/2)th power(m>1) of a value that is obtained by adding one to a square of a productof a change in a value of the evaluation function and an inversetemperature.
 4. The optimization device according to claim 1, whereinthe first transition probability distribution is a transitionprobability distribution represented by an exponential function of avalue obtained by multiplying by minus one an m-th power (m>0, wherem≠1) of a product of a change in a value of the evaluation function andan inverse temperature.
 5. The optimization device according to claim 1,wherein the processor is configured to calculate a value of theevaluation function each time the value of the state variable isupdated, and store a plurality of values in the memory in ascendingorder among calculated values of the evaluation function.
 6. Theoptimization device according to claim 1, wherein the processor isconfigured to output a value of the evaluation function at apredetermined sampling frequency.
 7. The optimization device accordingto claim 1, wherein the processor is configured to calculate, for eachof a plurality of annealing variables, a second Hamiltonian representedby a sum of a first product and a second product, the first productbeing a value obtained by multiplying a known first Hamiltonian with afirst value, the first value being a value obtained by subtracting avalue of the annealing variable from one, the second product being avalue obtained by multiplying a value of the evaluation function withthe each of the plurality of annealing variables, and output acalculation result of the calculating.
 8. The optimization deviceaccording to claim 7, wherein the processor is configured to select thepredetermined schedule from a plurality of schedules of increasing theannealing variable from zero to one.
 9. The optimization deviceaccording to claim 1, wherein the processor is further configured toperform a search for the minimum value by a replica exchange method, thereplica exchange method including: setting a different value of atemperature parameter for each of a plurality of replicas; andexchanging values of the temperature parameter among the plurality ofreplicas according to the set value of the temperature parameter and anexchange probability based on the transition probability distribution.10. The optimization device according to claim 1, wherein the processoris configured to: change the value of the parameter included in thetransition probability distribution; calculate an evaluation value withrespect to a value of the evaluation function at a value of eachparameter in a calculation time shorter than a calculation time whensearch for the minimum value of the evaluation function is performed;and output a calculation result.
 11. A method implemented by a computerfor controlling an optimization processing, the method comprising:obtaining, by a processor circuit of the computer, from a memory,setting information indicating an optimization method to be used, thememory being configured to store a value of a state variable included inan evaluation function obtained by converting a problem stored in thememory; selecting, by the processor circuit of the computer, based onthe setting information stored in the memory, the optimization method tobe used from among a plurality of optimization methods including asimulated annealing (SA) method and a replica exchange method; andperforming, by the processor circuit of the computer, the selectedoptimization method on the evaluation function by repeating a process ofupdating the value of the state variable by the selected optimizationmethod, the updating of the value of state variable being performed inaccordance with a first transition probability distribution that isrepresented by a first function, the first function being a functionconfigured to normalize to one a sum of probabilities of transition froma current state to each of a plurality of different states, the currentstate being represented by a current value of the state variable, thefirst transition probability distribution having a feature that atransition probability in the first transition probability distributionis larger than a transition probability in a Boltzmann distribution as achange in the value of the evaluation function due to a change in thevalue of the state variable is positively larger.
 12. A non-transitorycomputer-readable storage medium for storing a program for controllingan optimization processing, the program being configured to cause aprocessor circuit to perform processing, the processing comprising:obtaining, by the processor circuit, from a memory, setting informationindicating an optimization method to be used, the memory beingconfigured to store a value of a state variable included in anevaluation function obtained by converting a problem stored in a memory;and selecting, by the processor circuit of the computer, based on thesetting information stored in the memory, the optimization method to beused from among a plurality of optimization methods including asimulated annealing (SA) method and a replica exchange method; andperforming, by the processor circuit of the computer, the selectedoptimization method on the evaluation function by repeating a process ofupdating the value of the state variable by the selected optimizationmethod, the updating of the value of state variable being performed inaccordance with a first transition probability distribution that isrepresented by a first function, the first function being a functionconfigured to normalize to one a sum of probabilities of transition froma current state to each of a plurality of different states, the currentstate being represented by a current value of the state variable, thefirst transition probability distribution having a feature that atransition probability in the first transition probability distributionis larger than a transition probability in a Boltzmann distribution as achange in the value of the evaluation function due to a change in thevalue of the state variable is positively larger.